Numerical studies on Boussinesq-type equations via a split-step Fourier method

This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, redistribution , reselling , loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. Boussinesq-type nonlinear wave equations with dispersive terms are solved via split-step Fourier methods. We decompose the equations into linear and nonlinear parts, then solve them orderly. The linear part can be projected into phase space by a Fourier transformation, and resulting in a variable separable ordinary differential system, which can be integrated exactly. Next, by an invert Fourier transformation, the classical explicit fourth-order Runge–Kutta method is adopted to solve the nonlinear subproblem. To examine the numerical accuracy and efficiency of the method, we compare the numerical solutions with exact solitary wave solutions. Additionally, various initial-value problems for all the listed Boussinesq-type system are studied numerically. In the study, we can observe that sech 2-type waves for KdV-BBM system will split into several solitons, which is a very interesting physical phenomenon. The interaction between solitons, including overtaking and head-on collisions, is also simulated.